Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$. Along with this Hilbert space, I can also define raising/lowering and projection operators: $$ \begin{align} T_+^g |k\rangle = & \delta_{g,k} |k\rangle \\ T_-^g |k\rangle = & \delta_{g^{-1},k} |k\rangle \\ L_+^g |k\rangle = & |gk\rangle \\ L_-^g |k\rangle =&|kg^{-1}\rangle \end{align} $$ which may look familiar from quantum double models for topological order in 2+1D.

I could alternatively work in the representation basis, related to the group element basis by the Fourier transform: $$ |\mu,a,b\rangle = \sqrt{\frac{n_\mu}{|G|}} \sum_{g\in G}\Gamma^{ab}_\mu(g)|g\rangle $$ where $\mu$ labels irreps of the group, $n_\mu$ is the dimension of $\mu$, and $\Gamma^{ab}_\mu$ are the irrep matrices, with $a,b = 1,\dots,n_\mu$.

My question is the following: suppose I consider instead the group $G \times G$ with basis $\{|\mu i j\rangle|\nu k l\rangle\}$. Does there exist a projection operator which projects onto the subspace where irreps are tied together i.e. take the form $\{|\mu i j\rangle|\mu k l\rangle\}$?

I know the orthogonality relation $$ \sum_{g\in (G)_{cj}} |C| \chi_\mu(C) \bar{\chi}_\nu(C) = |G| \delta_{\mu,\nu} $$ where $(G)_{cj}$ is the set of conjugacy classes $C$ of $G$. This seems to do exactly what I want but I don't see that there's a way to express this in terms of the operators $L_{\pm}, T_{\pm}$.

In other words, I'd like to find an operator $P$ such that $$ \begin{equation} P |\mu i j\rangle |\nu k l\rangle = \delta_{\mu,\nu} \sum_{a,b}\sum_{c,d} A_{ab}^{ij}B_{cd}^{kl} |\mu a b\rangle|\mu cd\rangle \end{equation} $$ in terms of the raising and lowering operators, where I am allowing for the fact that there may be some mixing between states within the same irrep (if it has dimension >1).

Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$.

I could alternatively work in the representation basis, related to the group element basis by the Fourier transform: $$ |\mu,a,b\rangle = \sqrt{\frac{n_\mu}{|G|}} \sum_{g\in G}\Gamma^{ab}_\mu(g)|g\rangle $$ where $\mu$ labels irreps of the group, $n_\mu$ is the dimension of $\mu$, and $\Gamma^{ab}_\mu$ are the irrep matrices, with $a,b = 1,\dots,n_\mu$.

My question is the following: suppose I consider instead the group $G \times G$ with basis $\{|\mu i j\rangle|\nu k l\rangle\}$. Does there exist a projection operator which projects onto the subspace where irreps are tied together i.e. take the form $\{|\mu i j\rangle|\mu k l\rangle\}$?

In other words, I'd like to find an operator $P$ such that $$ \begin{equation} P |\mu i j\rangle |\nu k l\rangle = \delta_{\mu,\nu} \sum_{a,b}\sum_{c,d} A_{ab}^{ij}B_{cd}^{kl} |\mu a b\rangle|\mu cd\rangle \end{equation} $$ where I am allowing for the fact that there may be some mixing between states within the same irrep (if it has dimension >1).

Suppose I have a finite group G. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g>: g \in G \}$$. Along with this Hilbert space, I can also define raising/lowering and projection operators: $$ \begin{align} T_+^g |k> = & \delta_{g,k} |k> \\ T_-^g |k> = & \delta_{g^{-1},k} |k> \\ L_+^g |k> = & |gk> \\ L_-^g |k> =&|kg^{-1}> \end{align} $$ which may look familiar from quantum double models for topological order in 2+1D.

I could alternatively work in the representation basis, related to the group element basis by the Fourier transform: $$ |\mu,a,b> = \sqrt{\frac{n_\mu}{|G|}} \sum_{g\in G}\Gamma^{ab}_\mu(g)|g> $$ where $\mu$ labels irreps of the group, $n_\mu$ is the dimension of $\mu$, and $\Gamma^{ab}_\mu$ are the irrep matrices, with $a,b = 1,\dots,n_\mu$.

My question is the following: suppose I consider instead the group $G \times G$ with basis $\{|\mu i j>|\nu k l>\}$. Does there exist a projection operator which projects onto the subspace where irreps are tied together i.e. take the form $\{|\mu i j>|\mu k l>\}$?

I know the orthogonality relation $$ \sum_{g\in (G)_{cj}} |C| \chi_\mu(C) \bar{\chi}_\nu(C) = |G| \delta_{\mu,\nu} $$ where $(G)_{cj}$ is the set of conjugacy classes $C$ of $G$. This seems to do exactly what I want but I don't see that there's a way to express this in terms of the operators $L_{\pm}, T_{\pm}$.

In other words, I'd like to find an operator $P$ such that $$ \begin{equation} P |\mu i j> |\nu k l> = \delta_{\mu,\nu} \sum_{a,b}\sum_{c,d} A_{ab}^{ij}B_{cd}^{kl} |\mu a b>|\mu cd> \end{equation} $$ in terms of the raising and lowering operators, where I am allowing for the fact that there may be some mixing between states within the same irrep (if it has dimension >1).

Suppose I have a finite group $G$. With this group, I can associate an ortho-normal Hilbert space spanned by elements of the group $$\mathcal{H} = \{|g\rangle: g \in G \}$$. Along with this Hilbert space, I can also define raising/lowering and projection operators: $$ \begin{align} T_+^g |k\rangle = & \delta_{g,k} |k\rangle \\ T_-^g |k\rangle = & \delta_{g^{-1},k} |k\rangle \\ L_+^g |k\rangle = & |gk\rangle \\ L_-^g |k\rangle =&|kg^{-1}\rangle \end{align} $$ which may look familiar from quantum double models for topological order in 2+1D.

I could alternatively work in the representation basis, related to the group element basis by the Fourier transform: $$ |\mu,a,b\rangle = \sqrt{\frac{n_\mu}{|G|}} \sum_{g\in G}\Gamma^{ab}_\mu(g)|g\rangle $$ where $\mu$ labels irreps of the group, $n_\mu$ is the dimension of $\mu$, and $\Gamma^{ab}_\mu$ are the irrep matrices, with $a,b = 1,\dots,n_\mu$.

My question is the following: suppose I consider instead the group $G \times G$ with basis $\{|\mu i j\rangle|\nu k l\rangle\}$. Does there exist a projection operator which projects onto the subspace where irreps are tied together i.e. take the form $\{|\mu i j\rangle|\mu k l\rangle\}$?

I know the orthogonality relation $$ \sum_{g\in (G)_{cj}} |C| \chi_\mu(C) \bar{\chi}_\nu(C) = |G| \delta_{\mu,\nu} $$ where $(G)_{cj}$ is the set of conjugacy classes $C$ of $G$. This seems to do exactly what I want but I don't see that there's a way to express this in terms of the operators $L_{\pm}, T_{\pm}$.

In other words, I'd like to find an operator $P$ such that $$ \begin{equation} P |\mu i j\rangle |\nu k l\rangle = \delta_{\mu,\nu} \sum_{a,b}\sum_{c,d} A_{ab}^{ij}B_{cd}^{kl} |\mu a b\rangle|\mu cd\rangle \end{equation} $$ in terms of the raising and lowering operators, where I am allowing for the fact that there may be some mixing between states within the same irrep (if it has dimension >1).